Bring a large pot of water to a boil. Add the noodles and cook as the label directs. Drain in a colander and rinse under cold water, stirring the noodles with your hands. Set aside.
Mix the mushroom broth, sake, soy sauce, sugar and 1/4 teaspoon each salt and pepper in a large liquid measuring cup; set aside. Heat a large nonstick skillet over high heat until very hot. Add 1 tablespoon vegetable oil and heat until shimmering. Season the chicken lightly with salt and pepper; add to the skillet and cook until browned on the bottom, about 2 minutes. Flip and cook until just cooked through, about 1 minute. Transfer to a bowl and set aside.
Add the remaining 1 tablespoon vegetable oil to the skillet, then add the cabbage, carrot and scallion whites. Cook, stirring, until the cabbage is wilted and the carrot is crisp-tender, about 2 minutes. Add the snow peas and garlic and stir-fry until the garlic is golden, 1 to 2 more minutes. Add the noodles and stir-fry until dry and evenly combined, about 30 seconds.
Add the mushroom broth mixture and chicken to the skillet and cook, tossing, until the liquid is reduced and the noodles are coated, 1 to 2 more minutes; season with salt and pepper. Divide the noodles and vegetables among bowls and top with the scallion greens.